Gelfond’s method for algebraic independence

Abstract:

This paper extends Gelfond’s method for algebraic independence to fields $K$ with transcendence type $\leqslant \tau$. The main results show that the elements of a transcendence basis for $K$ and at least two more numbers from a prescribed set are algebraically independent over $Q$. The theorems have a common hypothesis: $\{ {\alpha _1}, \ldots ,{\alpha _M}\} ,\{ {\beta _1}, \ldots ,{\beta _N}\}$ are sets of complex numbers, each of which is $Q$-linearly independent. THEOREM A. If $(2\tau - 1) < MN$, then at least two of the numbers ${\alpha _i},{\beta _j},\exp ({\alpha _i}{\beta _j}),1 \leqslant i \leqslant M,1 \leqslant j \leqslant N$, are algebraically dependent over $K$. THEOREM B. If $2\tau (M + N) \leqslant MN + M$, then at least two of the numbers ${\alpha _i},\exp ({\alpha _i},{\beta _j}),1 \leqslant i \leqslant M,1 \leqslant j \leqslant N$, are algebraically dependent over $K$. THEOREM C. If $2\tau (M + N) \leqslant MN$, then at least two of the numbers $1 \leqslant i \leqslant M,1 \leqslant j \leqslant N$, are algebraically dependent over $K$.